\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  -19.0 & +  5.00 x_{1} &   & -9.00 x_{3} & +  9.00 x_{4} & +  7.00 x_{5} & -7.00 x_{6} & +  5.00 x_{7} & +  8.00 x_{8}\\
 x_{10}   &  -6.0 & +  7.00 x_{1} & -9.00 x_{2} & +  9.00 x_{3} & -4.00 x_{4} & -7.00 x_{5} & + 10.00 x_{6} & -1.00 x_{7} & -7.00 x_{8}\\
 x_{11}   &  22.0 & +  7.00 x_{1} &   & -6.00 x_{3} & -10.00 x_{4} & +  6.00 x_{5} & -5.00 x_{6} & -3.00 x_{7} & -7.00 x_{8}\\
 x_{12}   &  21.0 & +  4.00 x_{1} & -8.00 x_{2} & -5.00 x_{3} & +  3.00 x_{4} & -6.00 x_{5} & -6.00 x_{6} & -6.00 x_{7} & -9.00 x_{8}\\
 x_{13}   &  31.0 & +  2.00 x_{1} & +  6.00 x_{2} & -1.00 x_{3} & -10.00 x_{4} & -2.00 x_{5} & -6.00 x_{6} & -2.00 x_{7} & -3.00 x_{8}\\
\hline
z    &  0.0 & +  5.00 x_{1} & -3.00 x_{2} & -3.00 x_{3} & +  3.00 x_{4} & +  4.00 x_{5} & -2.00 x_{6} & -5.00 x_{7} &   \\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{8}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,13$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & -5.00 y_{9} & -7.00 y_{10} & -7.00 y_{11} & -4.00 y_{12} & -2.00 y_{13}\\
 y_{2}   &  1.0  &   & +  9.00 y_{10} &   & +  8.00 y_{12} & -6.00 y_{13}\\
 y_{3}   &  1.0 & +  9.00 y_{9} & -9.00 y_{10} & +  6.00 y_{11} & +  5.00 y_{12} & +  1.00 y_{13}\\
 y_{4}   &  1.0 & -9.00 y_{9} & +  4.00 y_{10} & + 10.00 y_{11} & -3.00 y_{12} & + 10.00 y_{13}\\
 y_{5}   &  1.0 & -7.00 y_{9} & +  7.00 y_{10} & -6.00 y_{11} & +  6.00 y_{12} & +  2.00 y_{13}\\
 y_{6}   &  1.0 & +  7.00 y_{9} & -10.00 y_{10} & +  5.00 y_{11} & +  6.00 y_{12} & +  6.00 y_{13}\\
 y_{7}   &  1.0 & -5.00 y_{9} & +  1.00 y_{10} & +  3.00 y_{11} & +  6.00 y_{12} & +  2.00 y_{13}\\
 y_{8}   &  1.0 & -8.00 y_{9} & +  7.00 y_{10} & +  7.00 y_{11} & +  9.00 y_{12} & +  3.00 y_{13}\\
\hline
z    &  -0 & + 19.00 y_{9} & +  6.00 y_{10} & -22.00 y_{11} & -21.00 y_{12} & -31.00 y_{13}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 3 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{10}   &  0.0481927710843 & +  0.06 y_{4} & -0.11 y_{1} & -1.36 y_{11} & -0.25 y_{12} & -0.82 y_{13}\\
 y_{2}   &  1.43373493976 & +  0.54 y_{4} & -0.98 y_{1} & -12.25 y_{11} & +  5.72 y_{12} & -13.37 y_{13}\\
 y_{3}   &  1.75903614458 & -1.30 y_{4} & +  0.54 y_{1} & + 22.81 y_{11} & +  3.27 y_{12} & + 15.10 y_{13}\\
 y_{9}   &  0.132530120482 & -0.08 y_{4} & -0.05 y_{1} & +  0.51 y_{11} & -0.45 y_{12} & +  0.75 y_{13}\\
 y_{5}   &  0.409638554217 & +  1.01 y_{4} & -0.42 y_{1} & -19.07 y_{11} & +  7.35 y_{12} & -8.96 y_{13}\\
 y_{6}   &  1.44578313253 & -1.19 y_{4} & +  0.75 y_{1} & + 22.16 y_{11} & +  5.41 y_{12} & + 19.42 y_{13}\\
 y_{7}   &  0.385542168675 & +  0.48 y_{4} & +  0.13 y_{1} & -0.89 y_{11} & +  7.98 y_{12} & -2.55 y_{13}\\
 y_{8}   &  0.277108433735 & +  1.10 y_{4} & -0.37 y_{1} & -6.58 y_{11} & + 10.80 y_{12} & -8.71 y_{13}\\
\hline
z    &  2.80722891566 & -1.24 y_{4} & -1.57 y_{1} & -20.55 y_{11} & -30.99 y_{12} & -21.72 y_{13}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  1.24096385542 & -0.06 x_{10} & -0.54 x_{2} & +  1.30 x_{3} & +  0.08 x_{9} & -1.01 x_{5} & +  1.19 x_{6} & -0.48 x_{7} & -1.10 x_{8}\\
 x_{1}   &  1.56626506024 & +  0.11 x_{10} & +  0.98 x_{2} & -0.54 x_{3} & +  0.05 x_{9} & +  0.42 x_{5} & -0.75 x_{6} & -0.13 x_{7} & +  0.37 x_{8}\\
 x_{11}   &  20.5542168675 & +  1.36 x_{10} & + 12.25 x_{2} & -22.81 x_{3} & -0.51 x_{9} & + 19.07 x_{5} & -22.16 x_{6} & +  0.89 x_{7} & +  6.58 x_{8}\\
 x_{12}   &  30.9879518072 & +  0.25 x_{10} & -5.72 x_{2} & -3.27 x_{3} & +  0.45 x_{9} & -7.35 x_{5} & -5.41 x_{6} & -7.98 x_{7} & -10.80 x_{8}\\
 x_{13}   &  21.7228915663 & +  0.82 x_{10} & + 13.37 x_{2} & -15.10 x_{3} & -0.75 x_{9} & +  8.96 x_{5} & -19.42 x_{6} & +  2.55 x_{7} & +  8.71 x_{8}\\
\hline
z    &  -2.80722891566 & -0.05 x_{10} & -1.43 x_{2} & -1.76 x_{3} & -0.13 x_{9} & -0.41 x_{5} & -1.45 x_{6} & -0.39 x_{7} & -0.28 x_{8}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  1.24096385542 & -0.06 x_{10} & -0.54 x_{2} & +  1.30 x_{3} & +  0.08 x_{9} & -1.01 x_{5} & +  1.19 x_{6} & -0.48 x_{7} & -1.10 x_{8}\\
 x_{1}   &  1.56626506024 & +  0.11 x_{10} & +  0.98 x_{2} & -0.54 x_{3} & +  0.05 x_{9} & +  0.42 x_{5} & -0.75 x_{6} & -0.13 x_{7} & +  0.37 x_{8}\\
 x_{11}   &  20.5542168675 & +  1.36 x_{10} & + 12.25 x_{2} & -22.81 x_{3} & -0.51 x_{9} & + 19.07 x_{5} & -22.16 x_{6} & +  0.89 x_{7} & +  6.58 x_{8}\\
 x_{12}   &  30.9879518072 & +  0.25 x_{10} & -5.72 x_{2} & -3.27 x_{3} & +  0.45 x_{9} & -7.35 x_{5} & -5.41 x_{6} & -7.98 x_{7} & -10.80 x_{8}\\
 x_{13}   &  21.7228915663 & +  0.82 x_{10} & + 13.37 x_{2} & -15.10 x_{3} & -0.75 x_{9} & +  8.96 x_{5} & -19.42 x_{6} & +  2.55 x_{7} & +  8.71 x_{8}\\
\hline
z    &  11.5542168675 & +  0.36 x_{10} & +  0.25 x_{2} & -1.81 x_{3} & +  0.49 x_{9} & +  3.07 x_{5} & -2.16 x_{6} & -7.11 x_{7} & -1.42 x_{8}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  1.24096385542 & -0.06 x_{10} & -0.54 x_{2} & +  1.30 x_{3} & +  0.08 x_{9} & -1.01 x_{5} & +  1.19 x_{6} & -0.48 x_{7} & -1.10 x_{8}\\
 x_{1}   &  1.56626506024 & +  0.11 x_{10} & +  0.98 x_{2} & -0.54 x_{3} & +  0.05 x_{9} & +  0.42 x_{5} & -0.75 x_{6} & -0.13 x_{7} & +  0.37 x_{8}\\
 x_{11}   &  20.5542168675 & +  1.36 x_{10} & + 12.25 x_{2} & -22.81 x_{3} & -0.51 x_{9} & + 19.07 x_{5} & -22.16 x_{6} & +  0.89 x_{7} & +  6.58 x_{8}\\
 x_{12}   &  30.9879518072 & +  0.25 x_{10} & -5.72 x_{2} & -3.27 x_{3} & +  0.45 x_{9} & -7.35 x_{5} & -5.41 x_{6} & -7.98 x_{7} & -10.80 x_{8}\\
 x_{13}   &  21.7228915663 & +  0.82 x_{10} & + 13.37 x_{2} & -15.10 x_{3} & -0.75 x_{9} & +  8.96 x_{5} & -19.42 x_{6} & +  2.55 x_{7} & +  8.71 x_{8}\\
\hline
z    &  11.5542168675 & +  0.36 x_{10} & +  0.25 x_{2} & -1.81 x_{3} & +  0.49 x_{9} & +  3.07 x_{5} & -2.16 x_{6} & -7.11 x_{7} & -1.42 x_{8}\\
\end{array}\]


 $ x_{2} $ enters and $ x_{4} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{2}   &  2.28888888889 & -0.11 x_{10} & -1.84 x_{4} & +  2.40 x_{3} & +  0.16 x_{9} & -1.87 x_{5} & +  2.20 x_{6} & -0.89 x_{7} & -2.02 x_{8}\\
 x_{1}   &  3.8  &   & -1.80 x_{4} & +  1.80 x_{3} & +  0.20 x_{9} & -1.40 x_{5} & +  1.40 x_{6} & -1.00 x_{7} & -1.60 x_{8}\\
 x_{11}   &  48.6  &   & -22.60 x_{4} & +  6.60 x_{3} & +  1.40 x_{9} & -3.80 x_{5} & +  4.80 x_{6} & -10.00 x_{7} & -18.20 x_{8}\\
 x_{12}   &  17.8888888889 & +  0.89 x_{10} & + 10.56 x_{4} & -17.00 x_{3} & -0.44 x_{9} & +  3.33 x_{5} & -18.00 x_{6} & -2.89 x_{7} & +  0.78 x_{8}\\
 x_{13}   &  52.3333333333 & -0.67 x_{10} & -24.67 x_{4} & + 17.00 x_{3} & +  1.33 x_{9} & -16.00 x_{5} & + 10.00 x_{6} & -9.33 x_{7} & -18.33 x_{8}\\
\hline
z    &  12.1333333333 & +  0.33 x_{10} & -0.47 x_{4} & -1.20 x_{3} & +  0.53 x_{9} & +  2.60 x_{5} & -1.60 x_{6} & -7.33 x_{7} & -1.93 x_{8}\\
\end{array}\]


 $ x_{5} $ enters and $ x_{2} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  1.22619047619 & -0.06 x_{10} & -0.99 x_{4} & +  1.29 x_{3} & +  0.08 x_{9} & -0.54 x_{2} & +  1.18 x_{6} & -0.48 x_{7} & -1.08 x_{8}\\
 x_{1}   &  2.08333333333 & +  0.08 x_{10} & -0.42 x_{4} &   & +  0.08 x_{9} & +  0.75 x_{2} & -0.25 x_{6} & -0.33 x_{7} & -0.08 x_{8}\\
 x_{11}   &  43.9404761905 & +  0.23 x_{10} & -18.85 x_{4} & +  1.71 x_{3} & +  1.08 x_{9} & +  2.04 x_{2} & +  0.32 x_{6} & -8.19 x_{7} & -14.08 x_{8}\\
 x_{12}   &  21.9761904762 & +  0.69 x_{10} & +  7.26 x_{4} & -12.71 x_{3} & -0.17 x_{9} & -1.79 x_{2} & -14.07 x_{6} & -4.48 x_{7} & -2.83 x_{8}\\
 x_{13}   &  32.7142857143 & +  0.29 x_{10} & -8.86 x_{4} & -3.57 x_{3} & -0.00 x_{9} & +  8.57 x_{2} & -8.86 x_{6} & -1.71 x_{7} & -1.00 x_{8}\\
\hline
z    &  15.3214285714 & +  0.18 x_{10} & -3.04 x_{4} & +  2.14 x_{3} & +  0.75 x_{9} & -1.39 x_{2} & +  1.46 x_{6} & -8.57 x_{7} & -4.75 x_{8}\\
\end{array}\]


 $ x_{3} $ enters and $ x_{12} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  3.44850187266 & +  0.01 x_{10} & -0.25 x_{4} & -0.10 x_{12} & +  0.07 x_{9} & -0.72 x_{2} & -0.24 x_{6} & -0.93 x_{7} & -1.37 x_{8}\\
 x_{1}   &  2.08333333333 & +  0.08 x_{10} & -0.42 x_{4} &   & +  0.08 x_{9} & +  0.75 x_{2} & -0.25 x_{6} & -0.33 x_{7} & -0.08 x_{8}\\
 x_{11}   &  46.9035580524 & +  0.32 x_{10} & -17.87 x_{4} & -0.13 x_{12} & +  1.06 x_{9} & +  1.79 x_{2} & -1.58 x_{6} & -8.79 x_{7} & -14.47 x_{8}\\
 x_{3}   &  1.72846441948 & +  0.05 x_{10} & +  0.57 x_{4} & -0.08 x_{12} & -0.01 x_{9} & -0.14 x_{2} & -1.11 x_{6} & -0.35 x_{7} & -0.22 x_{8}\\
 x_{13}   &  26.5411985019 & +  0.09 x_{10} & -10.90 x_{4} & +  0.28 x_{12} & +  0.05 x_{9} & +  9.07 x_{2} & -4.90 x_{6} & -0.46 x_{7} & -0.20 x_{8}\\
\hline
z    &  19.0252808989 & +  0.29 x_{10} & -1.81 x_{4} & -0.17 x_{12} & +  0.72 x_{9} & -1.69 x_{2} & -0.91 x_{6} & -9.33 x_{7} & -5.23 x_{8}\\
\end{array}\]


 $ x_{9} $ enters and $ x_{3} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  12.2142857143 & +  0.29 x_{10} & +  2.64 x_{4} & -0.50 x_{12} & -5.07 x_{3} & -1.43 x_{2} & -5.86 x_{6} & -2.71 x_{7} & -2.50 x_{8}\\
 x_{1}   &  13.0714285714 & +  0.43 x_{10} & +  3.21 x_{4} & -0.50 x_{12} & -6.36 x_{3} & -0.14 x_{2} & -7.29 x_{6} & -2.57 x_{7} & -1.50 x_{8}\\
 x_{11}   &  186.785714286 & +  4.71 x_{10} & + 28.36 x_{4} & -6.50 x_{12} & -80.93 x_{3} & -9.57 x_{2} & -91.14 x_{6} & -37.29 x_{7} & -32.50 x_{8}\\
 x_{9}   &  131.857142857 & +  4.14 x_{10} & + 43.57 x_{4} & -6.00 x_{12} & -76.29 x_{3} & -10.71 x_{2} & -84.43 x_{6} & -26.86 x_{7} & -17.00 x_{8}\\
 x_{13}   &  32.7142857143 & +  0.29 x_{10} & -8.86 x_{4} & +  0.00 x_{12} & -3.57 x_{3} & +  8.57 x_{2} & -8.86 x_{6} & -1.71 x_{7} & -1.00 x_{8}\\
\hline
z    &  114.214285714 & +  3.29 x_{10} & + 29.64 x_{4} & -4.50 x_{12} & -55.07 x_{3} & -9.43 x_{2} & -61.86 x_{6} & -28.71 x_{7} & -17.50 x_{8}\\
\end{array}\]


 $ x_{4} $ enters and $ x_{13} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  21.9758064516 & +  0.37 x_{10} & -0.30 x_{13} & -0.50 x_{12} & -6.14 x_{3} & +  1.13 x_{2} & -8.50 x_{6} & -3.23 x_{7} & -2.80 x_{8}\\
 x_{1}   &  24.9435483871 & +  0.53 x_{10} & -0.36 x_{13} & -0.50 x_{12} & -7.65 x_{3} & +  2.97 x_{2} & -10.50 x_{6} & -3.19 x_{7} & -1.86 x_{8}\\
 x_{11}   &  291.524193548 & +  5.63 x_{10} & -3.20 x_{13} & -6.50 x_{12} & -92.36 x_{3} & + 17.87 x_{2} & -119.50 x_{6} & -42.77 x_{7} & -35.70 x_{8}\\
 x_{9}   &  292.790322581 & +  5.55 x_{10} & -4.92 x_{13} & -6.00 x_{12} & -93.85 x_{3} & + 31.45 x_{2} & -128.00 x_{6} & -35.29 x_{7} & -21.92 x_{8}\\
 x_{4}   &  3.6935483871 & +  0.03 x_{10} & -0.11 x_{13} & +  0.00 x_{12} & -0.40 x_{3} & +  0.97 x_{2} & -1.00 x_{6} & -0.19 x_{7} & -0.11 x_{8}\\
\hline
z    &  223.701612903 & +  4.24 x_{10} & -3.35 x_{13} & -4.50 x_{12} & -67.02 x_{3} & + 19.26 x_{2} & -91.50 x_{6} & -34.45 x_{7} & -20.85 x_{8}\\
\end{array}\]


 $ x_{2} $ enters and Unbounded Dictionary!


 $ x_{2} $ enters and Unbounded Dictionary!
\end{document}
